Worked Interview Design Case ============================ This is a whiteboard-level first pass, not a production prescription. Its purpose is to demonstrate traceable reasoning, estimates, and the tests needed to retire uncertainty. Prompt ------ Outline a compact diode-pumped Nd:YAG laser delivering at least 5 W continuous wave at 1064 nm with :math:`M^2\le1.2` and stable linear polarization. Explain the resonator, pump, thermal, coating, measurement, and tolerance decisions. 1. Restate requirements and assumptions --------------------------------------- State unknowns rather than hiding them. For a first calculation assume: .. list-table:: :header-rows: 1 :widths: 35 25 40 * - Item - First-pass value - Must later come from * - Pump wavelength and incident power - 808 nm, 20 W - Qualified diode spectrum over current, temperature, and life * - Pump transport efficiency - 0.90 - Coating, fiber, lens, and alignment budget * - Absorbed fraction in crystal - 0.90 - Doping, length, spectrum, temperature, and double-pass design * - Gain length - 10 mm - Selected crystal and absorption/thermal model * - Cavity length - 100 mm optical first pass - Package, mode size, FSR, and tolerance trade * - Mirrors - Plane high reflector and :math:`R=200\,\mathrm{mm}` output coupler - Coating availability and thermal-lens sweep * - Output-coupler transmission - 5% - Saturated-gain/output-coupling optimization * - Other round-trip power loss - 2% represented exponentially - Loss measurement and coating/scatter budget 2. Pump and efficiency estimate ------------------------------- From :eq:`interview-pump-absorption`, .. math:: P_{\rm abs}=(0.90)(20\,\mathrm W)(0.90)=16.2\,\mathrm W. The ideal Stokes efficiency is .. math:: \eta_{\rm Stokes}=\frac{808}{1064}=0.759, so at least :math:`24.1\%` of absorbed pump becomes quantum-defect heat before other losses. If a provisional absorbed-power threshold is 3 W and absorbed slope efficiency is 45%, .. math:: P_{\rm out}\simeq0.45(16.2-3.0)=5.94\,\mathrm W. That passes the paper requirement but has little system margin. The next model must split transport, absorption, overlap, internal loss, output coupling, and thermal roll-over rather than tune the single 45% number. If the practical heat fraction is provisionally 35%, .. math:: P_{\rm heat}\simeq0.35(16.2)=5.67\,\mathrm W. This number drives the first thermal and mount model. 3. Cold-cavity eigenmode ------------------------ For the plane-concave cavity, .. math:: g_1=1,\qquad g_2=1-\frac{100}{200}=0.5,\qquad g_1g_2=0.5. The cold cavity is comfortably inside the ideal stability interval. With the waist at the plane mirror, .. math:: z_R=\sqrt{L(R-L)}=100\,\mathrm{mm}, .. math:: w_0=\sqrt{\frac{\lambda z_R}{\pi}} \simeq184\,\mu\mathrm m, and at the curved mirror .. math:: w(L)=w_0\sqrt{1+(L/z_R)^2}\simeq260\,\mu\mathrm m. These numbers are not final because the crystal's refractive surfaces and pump-dependent thermal lens belong in the round-trip matrix. They do establish the approximate pump-waist scale and optic clear-aperture requirement. 4. Threshold and circulating-power checks ------------------------------------------ Using :eq:`interview-threshold` with :math:`R_1=0.999`, :math:`R_2=0.95`, :math:`L_g=0.010\,\mathrm m`, and :math:`\mathcal L_i=0.02`, .. math:: g_{\rm th}\simeq \frac{\ln[1/(0.999\times0.95)]+0.02}{0.020} \simeq3.6\,\mathrm{m^{-1}}. The spectroscopy and inversion model must demonstrate this gain over the actual pumped volume. A 5-W output through :math:`T=0.05` implies roughly .. math:: P_{\rm circ}\simeq\frac{P_{\rm out}}{T}=100\,\mathrm W incident on the output coupler in this simplified travelling-wave power accounting. For a Gaussian radius of 260 µm, its on-axis irradiance is .. math:: I_{\rm pk}=\frac{2P_{\rm circ}}{\pi w^2} \simeq9.4\times10^8\,\mathrm{W/m^2} =94\,\mathrm{kW/cm^2}. Repeat this calculation at every optic and include standing-wave, defect, and transient enhancement before selecting coatings. 5. Pump overlap and thermal sweep --------------------------------- Start with an approximately 220-µm pump radius near the gain region so the pump slightly exceeds the cold TEM00 radius. Then calculate the full longitudinal absorbed-pump distribution and overlap integral in :eq:`interview-overlap`. Insert the gain medium and a variable thermal lens into the cavity matrix. Sweep from cold through the worst credible hot dioptric power, including tolerances. For every point record: * stability trace and distance to both stability boundaries; * beam radius through the pumped volume and on every coating; * pump/mode overlap and expected higher-order-mode discrimination; * clear-aperture clipping loss; * waist location and external mode-matching change; and * mirror-tilt sensitivity. If the thermal sweep crosses a stability boundary, change resonator geometry; do not merely plan to align more carefully. 6. Polarization, feedback, and mechanics ---------------------------------------- Use a polarization-selective element only if the gain/crystal geometry does not provide sufficient stable polarization. Budget the insertion loss and thermal depolarization. Tilt or wedge transmissive intracavity optics so ghosts cannot form a parasitic cavity, while tracking the astigmatism they introduce. Mount the crystal with a modeled and repeatable thermal interface, allow differential expansion, and avoid stress concentrations. Choose mirror mounts whose angular drift and resonances fit the alignment-sensitivity budget. Add isolation or slight angle to prevent output-path feedback into the resonator and pump diode where permitted by system requirements. 7. Verification plan -------------------- .. list-table:: :header-rows: 1 :widths: 31 36 33 * - Requirement or risk - Test - Pass evidence * - Output and efficiency - Incident and absorbed pump; output-versus-pump at stabilized temperatures - At least 5 W with declared efficiency basis and roll-over margin * - TEM00 beam quality - Two-axis second-moment caustic fit - :math:`M_x^2,M_y^2\le1.2` with residuals and uncertainty * - Polarization - Analyzer sweep over power and temperature - Required extinction ratio without mode or power instability * - Thermal robustness - Pump steps with waist, mode, power, and coolant logging - No stability crossing; model updated from measured thermal lens * - Alignment tolerance - Controlled mirror perturbation and environmental test - Requirement retained within allocated angular/positional range * - Coating/damage margin - Irradiance budget plus qualified optic data and inspection - Required statistical margin at wavelength and exposure condition The beam-quality row must be converted into the full acceptance statement described in :doc:`beam_quality`; the coating row must use the local-exposure and qualification framework in :doc:`laser_damage`. 8. Optional Q-switched extension -------------------------------- If the same 5-W average output is instead delivered at 20 kHz in 10-ns pulses, .. math:: E_p=250\,\mu\mathrm J, \qquad P_{\rm peak}\simeq25\,\mathrm{kW}. At a 300-µm :math:`1/e^2` radius, the Gaussian on-axis fluence is .. math:: F_{\rm pk}\simeq\frac{2E_p}{\pi w^2} \simeq0.177\,\mathrm{J/cm^2}, and peak irradiance is about :math:`17.7\,\mathrm{MW/cm^2}`. This immediately changes the Q-switch, coating, bulk-damage, nonlinear, detector, and beam-dump requirements even though average power is unchanged. What makes this a strong interview answer ----------------------------------------- It produces numbers quickly but labels them as assumptions, connects optical and thermal models, treats the thermal lens as a range, checks intracavity rather than output power alone, and ends with measurements capable of disproving the model.